Hamiltonian Definition and 901 Threads

I am attaching an image from David J. Griffith's "Introduction to Quantum Mechanics; Second Edition" page 205. In the scenario described (the Hamiltonian treats the two particles identically) it follows that $$PH = H, HP = H$$ and so $$HP=PH.$$ My question is: what are the necessary and...

Is Hamiltonian mechanics a mathematical generalization of Newtonian mechanics or is it explaining some fundamental relationship that has a meaning that extends into our nature ? I guess my question is what would led William Rowan Hamilton to come up with his type of mechanics or anything...

Hello! This is probably a silly question (I am sure I am missing something basic), but I am not sure I understand how a Hamiltonian can be a scalar and allow transitions between states with different angular momentum at the same time. Electromagnetic induced transitions are usually represented...

Hi, I have a question which asks me to use the generalised Ehrenfest Theorem to find expressions for ##\frac {d<Sx>} {dt}## and ##\frac {d<Sy>} {dt}## - I have worked out <Sx> and <Sy> earlier in the question. Since the generalised Ehrenfest Theorem takes the form...

In an Introduction to Quantum Mechanics by Griffiths (pg. 180), he claims that "P and H are compatible observables, and hence we can find a complete set of functions that are simultaneous eigenstates of both. That is to say, we can find solutions to the Schrodinger equation that are either...

Is the graph Hamilton and Eulerian? The website says the graph is Hamilton and Eulerian but I think it's wrong. Ref: https://scanftree.com/Graph-Theory/Eulerian-and-Hamiltonian-Graphs There is no path that covers all paths only once. Any help? I think the graph is drawn wrongly.

$$<f|\hat H g> = \int_{-\infty}^{\infty} f^*\Big(-\frac{\hbar}{2m} \frac{d^2}{dx^2} + V(x) \Big) g dx$$ Integrating (twice) by parts and assuming the potential term is real (AKA ##V(x) = V^*(x)##) we get $$<f|\hat H g> = -\frac{\hbar}{2m} \Big( f^* \frac{dg}{dx}|_{-\infty}^{\infty} -...

For the case when ##B=0## I get: $$Z = \sum_{n_i = 0,1} e^{-\beta H(\{n_i\})} = \sum_{n_i = 0,1} e^{-\beta A \sum_i^N n_i} =\prod_i^N \sum_{n_i = 0,1} e^{-\beta A n_i} = [1+e^{-\beta A}]^N$$ For non-zero ##B## to first order the best I can get is: $$Z = \sum_{n_i = 0,1}...

I do not understand the following sentence (particularly, the concept of extra symmetry): 'If all ##\alpha^i## are the same, then there is extra symmetry and corresponding constants of motion'. OK so let's find the Lagrangian; we know it has to have the form: $$L(q, \dot q) = T(q, \dot q) -...

Why is it the case that when some operators commute with the Hamiltonian (let's say A and ), it implies A and B commute, but even when each angular momentum component commutes with the Hamiltonian, it does not imply each the angular momentum components commute with each other?

Alright my idea is that, in order to show that ##f_i(q_i, p_i)## is a constant of motion, it would suffice to show that the Hamiltonian is equal to a constant. Well, the Hamiltonian will be equal to a constant iff: $$f(q_1, q_2, ..., q_N, p_1, p_2,..., p_N) = \text{constant}$$ Which is what...

I need help with part d of this problem. I believe I completed the rest correctly, but am including them for context (a)Show that the hermitian conjugate of the hermitian conjugate of any operator ##\hat A## is itself, i.e. ##(\hat A^\dagger)^\dagger## (b)Consider an arbitrary operator ##\hat...

Homework Statement: In the case of the quantum harmonic oscillator in 3D , does the z-component of the angular momentum of a particle commute with the Hamiltonian? Does the fundamental state has a well defined value of L_z (variance = 0) ? If you said no , why? If you said yes , what is the...

I have a question relates to a 3 levels system. I have the Hamiltonian given by: H= Acos^2 bt(|1><2|+|2><1|)+Asin^2 bt(|2><3|+|3><2|) I have been asked to find that H has an eigenvector with zero eigenvalues at any time t

In quantum mechanics, the Eigenfunction resulting from the Hamiltonian of a free particle in 1D system is $$ \phi = \frac{e^{ikx} }{\sqrt{2\pi} } $$ We know that a function $$ f(x) $$ belongs to Hilbert space if it satisfies $$ \int_{-\infty}^{+\infty} |f(x)|^2 dx < \infty $$ But since the...

I have always seen this problem formulated in a well that goes from 0 to L I am confused how to use this boundary, as well as unsure of what a dimensionless hamiltonian is. This is as far as I have gotten

Hi I'm looking at Tong notes http://www.damtp.cam.ac.uk/user/tong/qhe/two.pdf deriving the Kubo Formula, section 2.2.3, page 54,I don't understand where the Hamiltonian comes from (eq 2.8). I tried a quick google but couldn't find anything. I'm not very familiar with EM Hamiltonians, any help/...

Hi! So this is my first homework ever of Hamiltonian dynamics and I am struggling with the understanding of the most basic concepts. My lecturer is following Saletan's and Deriglazov's and from what I have read and from my lectures, this is what I think I know. Please let me know if this is...

Hi! I recently came across a quantum mechanics problem involving a change of basis to a rotating basis. As part of the solution, I wanted to transform the Hamiltonian operator into the rotating basis. Since the new basis is rotating, the basis change operator is time-dependent. This led to a...

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Dear PF, I hope I've formulated my question understandable enough. Thank you for your time, Garli

Hello, I'm working on a project. I need to understand every equation in a paper. I need to calculate the spatial derivative of G (d/dR), a two-particle Hamiltonian. However, G is a function of P- the density matrix and P is a function of R. Is it a "special derivative"? Here is the attached...

I am given this Hamiltonian: And asked to diagonalize. I understand how we do such a Hamiltonian: But I don't understand how to deal with the extra term in my given Hamiltonian. Usually we use To get

I'm reading a book about analytical mechanics and in particular, in a chapter on hamiltonian Mechanics it says: "In the state space (...) the complete solutionbof the canonical equations is pictured as an infinite manifold of curves which fill (2n+1)-dimensional space. These curves never cross...

Please see this page and give me an advice. https://physics.stackexchange.com/questions/499269/simultanious-eigenstate-of-hubbard-hamiltonian-and-spin-operator-in-two-site-mod Known fact 1. If two operators ##A## and ##B## commute, ##[A,B]=0##, they have simultaneous eigenstates. That means...

In Newtonian mechanics, conservation laws of momentum and angular momentum for an isolated system follow from Newton's laws plus the assumption that all forces are central. This picture tells nothing about symmetries. In contrast, in Hamiltonian mechanics, conservation laws are tightly...

I am currently reading Griffiths Introduction to Quantum Mechanics, 2nd Edition. I am aware that, in light of considering potential functions independent of time, the Schrödinger equation has separable solutions and that these solutions are stationary states. I am also aware (If I stand correct)...

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Summary: I have found the Hamiltonian for the free particles and the electromagnetic field (##\mathbf{E}## - electric, ##\mathbf{B}## - magnetic) to be (non-relativistic !): ##H=\sum_i \frac{m \dot{r}_i^2}{2} + \int d^3 r \left(\frac{\epsilon_0}{2} E^2 + \frac{1}{2\mu_0} B^2\right)## (1)...

I have been reading a book on classical theoretical physics and it claims: -------------- If a Lagrange function depends on a continuous parameter ##\lambda##, then also the generalized momentum ##p_i = \frac{\partial L}{\partial\dot{q}_i}## depends on ##\lambda##, also the velocity...

In QM. Can you use the Hamiltonian (or kinetic plus potential energy) to describe the forces of nature that should supposedly use QFT or Lagrangian? I mean the fields have kinetic and potential energy components and what are the limitations in description and others if you only use them to...

Hello! I am stuck at the following question: Show that the wave function is an eigenfunction of the Hamiltonian if the two electrons do not interact, where the Hamiltonian is given as; the wave function and given as; and the energy and Born radius are given as: and I used this for ∇...

In the Schrodinger picture, the operators don't change with the time, but the states do. So, what happen if my hamiltonian depend on time? Should I use the others pictures in these cases?

It seems as though in general, the equations of motion are described with two equations which result from the definition of a Hamiltonian problem, where the problems are of the form: $$\dot p=-H_q(p,q), \dot q=H_p(p,q)$$ It is a little confusing to me how the equations of motion go from two...

Greg Bernhardt submitted a new blog post A Pure Hamiltonian Proof of the Maupertuis Principle Continue reading the Original Blog Post.

Hi, I hope this is in the right section. It's for EM which I guess is a relativistic theory but the question itself is not to do with any Lorentz transformations or anything similar. I'm reading through Jackson with my course for EM and I'm on the section where he is generalising the Hamiltonian...

I need to learn about Hamiltonian mechanics involving functional and functional derivative... Also, I need to learn about generalized real and imaginary Hamiltonian... I only learned the basics of Hamiltonian mechanics during undergrad and now those papers I read show very generalized version...

"Once, Dirac asked me whether I thought geometrically or algebraically? I said I did not know what he meant, could he tell me how he, himself, thought. He said his thinking was geometrical. I was taken aback by this because Dirac, with his transformation theory, represented for my generation the...

I want to exactly diagonalize the following Hamiltonian for ##10## number of sites and ##5## number of spinless fermions $$H = -t\sum_i^{L-1} \big[c_i^\dagger c_{i+1} - c_i c_{i+1}^\dagger\big] + V\sum_i^{L-1} n_i n_{i+1}$$ here ##L## is total number of sites, creation (##c^\dagger##) and...

I have been attempting to modify a symplectic integrator that I wrote a while ago. It works very well for "separable" hamiltonians, but I want to use it to simulate a double pendulum. I am using the Stormer-Verlet equation (3) from this source. From the article "Even order 2 follows from its...

Hi, I am really new in understanding second quantization formalism. Recently I am reading this journal: https://dash.harvard.edu/bitstream/handle/1/8403540/Simulation_of_Electronic_Structure.pdf?sequence=1&isAllowed=y In brief, the molecular Hamiltonian is written as $$\mathcal{H}=\sum_{ij}...

Well I think it is cool anyhow ;) Here is a dependency-free variable-order double pendulum simulation in about 120 lines of Python. Have you seen the equations of motion for this system? As usual, this is based on code that I have provided here, but trimmed for maximum impact. Can you see...

hi all! I’m trying to generalise the Caldeira-Leggett Hamiltonian (heat bath + particle) to the case of high velocities. Naturally, the multi-oscillator Hamiltonian needs to change and I have a gnawing suspicion that the multi-particle Hamiltonian is just the sum of single-particle hamiltonians...

Hi, could you please explain me, how to copmute the yellow parts? Thank you very much.

in the Lagrangian mechanics, we assumed that the Lagrangian is a function of space coordinates, time and the derivative of those space coordinates by time (velocity) L(q,dq/dt,t). to derive the Hamiltonian we used the Legendre transformation on L with respect to dq/dt and got H = p*(dq/dt) -...

I was looking at the following proof of Louviles theorem : we define a velocity field as V=(dpi/dt, dqi/dt). using Hamilton equations we find that div(V)=0. using continuity equation we find that the volume doesn't change. I couldn't figure out the following : 1- the whole point was to show...

Homework Statement I have the diatomic molecule hamiltonian given by: $$-\hbar^2/(2\mu)d^2/dr^2+\hbar^2\ell(\ell+1)/(2\mu r^2)+(1/4)K(r-d_0)^2$$ Now it's written in my solutions that if we put: $$K\equiv 2\mu \omega_0^2, \hbar^2\ell(\ell+1)/(2\mu d_0^4)\equiv \gamma_{\ell} K, r-d_0\equiv...

Hey, I have this chaotic system. It is a modified Hamiltonian Chaotic System and it is based on Henon-Heiles chaotic system. So I have this functions (as shown below). I want to know how can I make it as a discrete function. Like, how can I know the value for x dot and y dot. 1. Prefer to know...

Hi guys! I am starting to study Hubbard model with application in DFT and I have some doubts how to solve the Hubbard Hamiltonian. I have the DFT modeled to Hubbard, where the homogeneous Hamiltonian is $$ H = -t\sum_{\langle i,j \rangle}\sigma (\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma} +...

I know, $$ L=T-V \;\;\; \; \;\;\; [1]\;\;\; \; \;\;\; ( Lagrangian) $$ $$ H=T+V \;\;\; \; \;\;\;[2] \;\;\; \; \;\;\; (Hamiltonian)$$ and logically, this leads to the equation, $$ H - L= 2V \;\;\; \; \;\;\...